Problem: Determine how many solutions exist for the system of equations. ${9x-3y = 3}$ ${-3x+y = 8}$
Solution: Convert both equations to slope-intercept form: ${9x-3y = 3}$ $9x{-9x} - 3y = 3{-9x}$ $-3y = 3-9x$ $y = -1+3x$ ${y = 3x-1}$ ${-3x+y = 8}$ $-3x{+3x} + y = 8{+3x}$ $y = 8+3x$ ${y = 3x+8}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 3x-1}$ ${y = 3x+8}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.